Representations and Modules of Rota-Baxter Algebras

TL;DR

This paper provides a comprehensive study of the representation and module theory of Rota-Baxter algebras, including decompositions, categorical equivalences, and applications to coalgebras and tensor categories.

Contribution

It introduces new structures and equivalences in Rota-Baxter algebra representations, including regular-singular decompositions and categorical frameworks.

Findings

Decomposition of Rota-Baxter algebras and modules under quasi-idempotency

Equivalence between Rota-Baxter algebra representations and Rota-Baxter operator ring representations

Explicit constructions and algebraic Birkhoff factorization in coalgebra representations

Abstract

We give a broad study of representation and module theory of Rota-Baxter algebras. Regular-singular decompositions of Rota-Baxter algebras and Rota-Baxter modules are obtained under the condition of quasi-idempotency. Representations of an Rota-Baxter algebra are shown to be equivalent to the representations of the ring of Rota-Baxter operators whose categorical properties are obtained and explicit constructions are provided. Representations from coalgebras are investigated and their algebraic Birkhoff factorization is given. Representations of Rota-Baxter algebras in the tensor category context is also formulated.